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0028: Part 7, Singular Moduli and Dedekind eta function

Part 7: Four Singular Moduli and the Complete Elliptic Integral of the First Kind

by Tito Piezas III

Abstract: Formulas for four kinds of singular moduli will be given in terms of the Dedekind eta function η(τ) at arguments τ = 1/p√-m.However, at τ = (1+√-d)/2, these can also be used to compute the complete elliptic integral of the first kindK(kd).

I. Introduction

II. Four Singular Moduli

III. Complete Elliptic Integral of the First Kind

I. Introduction

In his second letter to Hardy, Ramanujan solved the equation involving the hypergeometric function2F1(a,b;c;z),

and {x, y} = {29, 2} is the fundamental solution to the Pell equation x2-210y2 = 1.There is a general formula for δ, and similar singular moduli, using the Dedekind eta function to be given here. (However, to express them as products of units as Ramanujan ingeniously did is another matter entirely.)

Ramanujan’s example is then p = 4, τ = (1/4)√-840, and its r4(τ) is easily calculated using the Dedekind eta function, though to express it as radicals ab initio is not so easy.In general, for m a positive integer, then α, β, γ, δ, are algebraic numbers.

III. Complete Elliptic Integral of the First Kind

The complete elliptic integral of the first kind K(k) is defined for 0 < k < 1 by,

or equivalently,

But it can also be expressed in terms of α, β, δ (missing the third moduli, γ) as,

where τ is now restricted to the form,

for positive dabove a bound, otherwise rp(τ) may be zero. For example, for p = 1, then at least d > 3, since for d = 3, it is well-known that r1(τ) = j(τ) = 0.But, say, for d = 163, then,

where,

and x is the real root of the cubic, x3-6x2+4x-2 = 0. Similarly, for p = 2, and d = 37,